We may extend \(F_k\)  to a positive operator-valued measure (POVM) \((F_k)_{k=0}^K\) , \(F_k \in \mathcal {A}_+\)  by setting \(F_0 := 1 - \sum _{k=1}^K F_k\)  such that \(\sum _{k=0}^K F_k = 1\) . By Naimark's theorem [1]}, [2]}, \(F\)  admits a dilation \(F = \tilde{v}^* \pi \tilde{v}\) , where \(\tilde{v}: \mathcal {H}_{\mathcal {A}} \rightarrow \mathcal {H}_{\tilde{\mathcal {A}}}\)  is a linear map and \((\pi _k)_{k=0}^K\) , \(\pi _k \in \mathcal {P}(\mathcal {H}_{\tilde{\mathcal {A}}})\)  a projection-valued measure (PVM). Consequently, \(\phi _\rho (a) = \tilde{\phi }_\rho (a) - E_0\ \mathrm {tr}[F_0 a]\) , where e.g. \(E_0 \propto \mathbb {1}\)  and
\(\tilde{\phi }_\rho (a)= \sum _{k=0}^K E_k\ \mathrm {tr}_{\mathcal {H}_{\mathcal {A}}}[\tilde{v}^*\pi _k\tilde{v}a]= \sum _{k=0}^K E_k\ \mathrm {tr}_{\mathcal {H}_{\mathcal {A}}}[\tilde{v}^*\pi _k \tilde{\Phi }_\rho (a)\tilde{v}]= \sum _{k=0}^K E_k\ \mathrm {tr}_{\mathcal {H}_{\tilde{\mathcal {A}}}}[\pi _k\tilde{\Phi }_\rho (a)]\; .\) 
